In 1841 they had in their household:
William 22, Elizabeth 18, May 16, Harriet 14, Sarah 7, and James 15. James is listed out of order. There are no relationships on the census.
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I think this is their arrival:
arrived New York June 16, 1845, on the James Moran from Liverpool
James Barrow, 59, a farmer
Sarah 39 wife
Betsey 20 daughter
William 26 son
Mary Ann 23 daughter
Hannah 13 daughter
Sarah 10 daughter
all English, all intend to stay in the US
[that is what Sarah's age said. It doesn't match William's birth and barely matches Mary Ann's]
There is a John Burrows, 19, on the same page of the manifest, but he is a blacksmith from Ireland. I didn't find another John Burrows on the manifest.
https://www.familysearch.org/ark:/61903/1:1:QVRQ-7TGT